Research house IHS says that they expect flexible OLED smartphone display shipments to reach 814,000 units in 2014. IHS sees a steady but 'modest' growth in the future as shipments will almost double each year to reach 8.3 million units in 2018.
In August 2013 IHS said that they expect flexible OLED sales to reach $21.9 million in 2013 and grow quickly to almost $100 million in 2014. The flexible OLED market will continue growing at a very fast rate as it will reach almost $4 billion by 2018 and almost $12 billion by 2020.
The chart above shows IHS' growth and revenue estimates from half a year ago. So it seems the company is now less optimistic about flexible displays. Of course the August numbers include all flexible OLEDs, not just mobile phone ones, so that may explain that.
DisplaySearch are much more aggressive in their estimates - they actually see flexible OLEDs taking up 40% of the total smartphone display market in 2018, up from 0.2% in 2013. From 2013 to 2015 the market will grow sixfold, according to displaysearch.